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Refined uniform estimates of oscillatory integrals and areas. (Russian) Zbl 0692.47015
Let f: \(R^ 2\to R\), \(\phi\) : \(R^ 2\to R\) and \[ I(\tau,f,\phi)=\int_{R^ 2}\phi \exp \{i\tau f\},\quad V(\epsilon,f,c,\phi,A)=\int_{R^ 2}\chi \phi, \] where \(\chi\) is the indicator of the set \(\{x\in A:\quad c-\epsilon \leq f(x)\leq c+\epsilon \}\) and A is an open set. In the paper some uniform two-term upper estimates for \(| I(\tau,f,\phi)|\) and \(| V(\epsilon,f,c,\phi,A)|\) are obtained.
Reviewer: Yu.M.Ryžov

MSC:
47A55 Perturbation theory of linear operators
26D10 Inequalities involving derivatives and differential and integral operators
58J40 Pseudodifferential and Fourier integral operators on manifolds
42B99 Harmonic analysis in several variables
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